Tikhonov regularization applied to the inverse problem of option pricing: convergence analysis and rates

Abstract

Abstract. This paper investigates the stable identification of local volatility surfaces σ(S, t) in the Black-Scholes/Dupire equation from market prices of European Vanilla options. Based on the properties of the parameter-to-solution mapping, which assigns option prices to given volatilities, we show stability and convergence of approximations gained by Tikhonov regularization. In case of a known term-structure of the volatility surface, in particular if the volatility is assumed to be constant in time (σ(S, T) = σ(S)), we prove convergence rates under simple smoothness and decay conditions on the true volatility. The convergence rate analysis sheds light onto the importance of an appropriate a-priori guess for the unknown volatility and the nature for the ill-posedness of the inverse problem, caused by smoothing properties and the nonlinearity of the direct problem. Finally, the theoretical results are illustrated by numerical experiments. 1.